Some higher-order multi-point boundary value problem at resonance

This paper deals with the existence of solutions for the following n th order multi-point boundary value problem at resonance x ( n ) ( t ) = f ( t , x ( t ) , x ′ ( t ) , … , x ( n - 1 ) ( t ) ) + e ( t ) , t ∈ ( 0 , 1 ) , x ( 0 ) = ∑ i = 1 m - 2 α i x ( ξ i ) , x ′ ( 0 ) = ⋯ = x ( n - 2 ) ( 0 ) = 0 , x ( 1 ) = x ( η ) , where f : [ 0 , 1 ] × R n ⟶ R is a continuous function, e ∈ L 1 [ 0 , 1 ] , α i ∈ R ( 1 ⩽ i ⩽ m - 2 ) , 0 < ξ 1 < ξ 2 < ⋯ < ξ m - 2 < 1 and 0 < η < 1 . An existence theorem is obtained by using the coincidence degree theory of Mawhin.